Cohomological Hall Algebras of Calabi-Yau 3-folds

“Calabi-Yau三折”是六维弯曲空间，在几何学和理论物理的弦理论中都是重要的。弦理论一直只在10维时空中定义，为了描述我们的4维时空(3个空间维度和1个时间维度)，人们需要将额外的6个维度包裹在一个非常小的Calabi-Yau 3折上。Calabi-Yau 3重的几何形状决定了我们的4维物理(粒子等)。通过将四维物理理论与Calabi-Yau三重空间联系起来(这在数学上还没有被理解)，弦理论家对Calabi-Yau三重空间做出了令人惊叹的猜想，这一区域被称为镜像对称性。“Donaldson-Thomas不变量”是对居住在Calabi-Yau三重X上的几何物体(相干轨道)进行计数的数字。相干轨道形成了一个“模空间”M，这是一个奇异空间，而DT不变量是通过在M上的一种不寻常的积分来定义的。在弦理论中，DT不变量是“BPS态的数目”，这是一种粒子。2008年，Pi和Yinan Song展示了如何在最一般的情况下定义DT不变量，并证明了它们随着X上的结构变形而变化的“穿墙公式”。这导致了对唐纳森-托马斯理论及其扩展的研究的爆炸性增长。2013年，PI证明了DT不变量可以解释为向量空间的维度。作为模空间M的一种奇异上同调，向量空间有一种困难的构造(上同调度量空间M的“形状”，例如，甜甜圈中的洞)。在弦论中，这些向量空间是“BPS态的矢量空间”，是与X有关的量子场论的一部分。几何(Kontsevich-Soibelman)和弦论(Harvey-Moore)中的一个由来已久的猜想是，这些向量空间上应该有一个乘法，使它们成为一个代数(像普通数一样具有加法和乘法的东西)--物理学文献中的“BPS态代数”，或数学文献中的“上同调霍尔代数(COHA)”。在物理学中，相乘来自两个粒子结合形成第三个粒子。2010年，Kontsevich-Soibelman为Calabi-Yau设计了一个玩具模型，名为“具有超势的箭袋”，证实了这个猜想。PI和Pavel Safronov在2015年的工作使人们能够定义模空间M的小区域上的乘法，但还不能定义整个空间上的乘法。在这个方案中，我们的目的是构造BPS态的向量空间上的乘法。为此，我们将用一种新的方法证明2013年以来在移辛导出的代数几何领域中PI的一个更一般的猜想，证明这个猜想使我们能够定义Calabi-Yau 3-折叠的Coha，这是无限维代数，DT不变量是这些代数片的维度。然后，我们可以用表示论来研究这些代数，例如，可以证明DT不变量是模形式的幂级数系数，这是数论中的一类特殊函数。我们将证明的猜想还具有其他非常重要的应用，我们在提案中探索了这些应用：*它给出了由Joyce和Borisv在2015年定义的8维Calabi-Yau 4重X的另一种构造，并证明了这些DT4不变量具有其他有用的性质，例如，它们在将X分成两部分时的行为。*它允许我们定义辛流形的“Fukaya范畴”的代数几何版本，这是镜像对称的关键。这个代数版本将比通常的版本更简单和更严格，并且不需要许多通常的限制性假设。*这种“Fukaya范畴”具有重要的应用，包括研究普通3维空间中的结，以及将结的不变量转化为称为拓扑量子场论的数学结构。

"Calabi-Yau 3-folds" are 6-dimensional curved spaces that are important in Geometry, and also in String Theory in Theoretical Physics. String Theory is consistently defined only in 10-dimensional spacetime, and in order to describe our 4-dimensional spacetime (3 space dimensions and 1 time dimension), one is required to wrap the additional 6 dimensions on a very small Calabi-Yau 3-fold. The geometry of the Calabi-Yau 3-fold determines our 4-dimensional physics (particles, etc). By associating a 4-dimensional physical theory to the Calabi-Yau 3-fold, which is not mathematically understood, String Theorists make amazing conjectures about Calabi-Yau 3-folds, an area known as Mirror Symmetry."Donaldson-Thomas invariants" are numbers counting geometric objects (coherent sheaves) living on a Calabi-Yau 3-fold X. The coherent sheaves form a "moduli space" M, a singular space, and DT invariants are defined by an unusual kind of integration over M. In String Theory, DT invariants are "numbers of BPS states", a kind of particle. In 2008, the PI and Yinan Song showed how to define DT invariants in the most general case, and proved they change by a "wall-crossing formula" as the structure on X deforms. This led to an explosion of research on Donaldson-Thomas theory and its extensions. In 2013, the PI showed DT invariants can be interpreted as dimensions of vector spaces. The vector spaces have a difficult construction as a kind of exotic cohomology of the moduli space M. (Cohomology measures the "shape" of a space M, e.g. the hole in a donut.) In String Theory, these vector spaces are "vector spaces of BPS states", part of the Quantum Field Theory associated to X.It is a long standing conjecture in Geometry (Kontsevich-Soibelman) and String Theory (Harvey-Moore) that these vector spaces should have a multiplication on them making them into an algebra (something with addition and multiplication, like ordinary numbers) - the "algebra of BPS states" in the Physics literature, or "Cohomological Hall Algebra (CoHA)" in the mathematics literature. In Physics, the multiplication comes from two particles joining to make a third particle. The conjecture was proved in 2010 by Kontsevich-Soibelman for "quivers with superpotential", a toy model for Calabi-Yau 3-folds. Work by the PI and Pavel Safronov in 2015 enables one to define the multiplication over small regions of the moduli space M, but not yet over the whole space.In this proposal, we aim to construct the multiplication on the vector space of BPS states. We will do this by proving a much more general conjecture of the PI from 2013 in the area of Shifted Symplectic Derived Algebraic Geometry, by a new method.Proving this conjecture enables us to define CoHAs for Calabi-Yau 3-folds, which are infinite-dimensional algebras, and DT invariants are dimensions of pieces of these algebras. We can then study these algebras using Representation Theory, e.g. it may be possible to show that DT invariants are power series coefficients of modular forms, a class of special functions in Number Theory.The conjecture we will prove also has other very important applications, which we explore in the proposal:* It gives an alternative construction of "DT4 invariants" of 8-dimensional Calabi-Yau 4-folds X, defined by Joyce and Borisov in 2015, and shows these DT4 invariants have additional useful properties, e.g. how they behave on cutting X into 2 pieces.* It allows us to define an algebraic geometry version of the "Fukaya category" of a symplectic manifold, which are key to Mirror Symmetry. This algebraic version will be simpler and more rigid than the usual version, and work without many of the usual restrictive assumptions.* This "Fukaya category" has important applications, including to the study of knots in ordinary 3-dimensional space, and to making invariants of knots into a mathematical structure called a Topological Quantum Field Theory.